all the regular polytopes in all dimensions have been enumerated (things become boring in 5 dimensions and higher), but it seems a similar enumeration of all convex uniform polytopes has not yet been accomplished. interesting new symmetries are known to become possible in 6, 7, and 8 dimensions.
are there interesting polytopes in 24 dimensions or thereabouts? certainly the Voronoi cell (cells? are all cells congruent?) of the Leech lattice, though I have no idea if it or its dual is uniform.
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